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Enumeration of binary trees and universal types
 Discrete Math. Theor. Comput. Sci
"... Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., heig ..."
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Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a given path length (sum of depths) are there? This question arose in the study of universal types of sequences. Two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the LempelZiv’78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, Tp, of given path length p (and also the number of distinct LempelZiv’78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to Tp ∼ 2 2p/(log 2 p)(1+O(log−2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrases in the LempelZiv’78 scheme) when a tree is selected randomly among all trees of given path length p. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well
On the joint path length distribution in random binary trees
 Stud. Appl. Math
"... During the 10th Seminar on Analysis of Algorithms, MSRI, Berkeley, June 2004, Knuth posed the problem of analyzing the left and the right path length in a random binary trees. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right p ..."
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During the 10th Seminar on Analysis of Algorithms, MSRI, Berkeley, June 2004, Knuth posed the problem of analyzing the left and the right path length in a random binary trees. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right path lengths. In this paper, we mostly focus on the asymptotic properties of the distribution of the difference between the left and the right path lengths. Among other things, we show that the Laplace transform of the appropriately normalized moment generating function of the path difference satisfies the first Painlevé transcendent. This is a nonlinear differential equation that has appeared in many modern applications, from nonlinear waves to random matrices. Surprisingly, we find out that the difference between path lengths is of the order n 5/4 where n is the number of nodes in the binary tree. This was also recently observed by Marckert and Janson. We present precise asymptotics of the distribution’s tails and moments. We shall also discuss the joint distribution of the left and right path lengths. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method.
Analysis of Algorithms (AofA): Part I: 1993  1998 ("Dagstuhl Period")
"... This is the rst installment of the Algorithmics Column dedicated to Analysis of Algorithms (AofA) that sometimes goes under the name AverageCase Analysis of Algorithms or Mathematical Analysis of Algorithms. The area of analysis of algorithms (at least, the way we understand it here) was born on ..."
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This is the rst installment of the Algorithmics Column dedicated to Analysis of Algorithms (AofA) that sometimes goes under the name AverageCase Analysis of Algorithms or Mathematical Analysis of Algorithms. The area of analysis of algorithms (at least, the way we understand it here) was born on July 27, 1963, when D. E. Knuth wrote his \Notes on Open Addressing". Since 1963 the eld has been undergoing substantial changes. We report here how it evolved since then. For a long time this area of research did not have a real \home". But in 1993 the rst seminar entirely devoted to analysis of algorithms took place in Dagstuhl, Germany. Since then seven seminars were organized, and in this column we briey summarize the rst three meetings held in Schloss Dagstuhl (thus \Dagstuhl Period") and discuss various scienti c activities that took place, describing some research problems, solutions, and open problems discussed during these meetings. In addition, we describe three special issues dedicated to these meetings.